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GALOIS REPRESENTATIONS in ARITHMETIC GEOMETRY II The GALOIS REPRESENTATIONS IN ARITHMETIC GEOMETRY II TAKESHI SAITO The ´etale cohomology of an algebraic variety defined over the rational number field Q gives rise to an -adic representation of the absolute Galois group GQ of Q. To study such an -adic representation, it is common to in- vestigate its restriction to the decomposition group at each prime number p. The purpose of this article is to survey our knowledge on the restrictions at various primes. When the variety has bad reduction at p, we find an interesting phenomenon called ramification. In this article we also report on recent discoveries on ramification. This article is closely related to but independent of [26]. 1. Local-Global relation in number theory 1.1. The Beta function and the Jacobi sums. Consider the Jacobi sum J(a, b)= χa(x)χb(1 − x), x∈Fp\{0,1} where χ is a multiplicative character of Fp, that is, a character of the mul- × tiplicative group Fp . This is an arithmetic analog of the Beta function 1 B(s, t)= xs−1(1 − x)t−1 dx. 0 Their formal similarity comes from the fact that the function x → xs is a multiplicative character of R, and the summation in the Jacobi sum is replaced by an integral in the Beta function. The Beta function and the Jacobi sums share a number of similar properties. The similarity is not superficial but is rooted in geometry. Let Cn be the n n algebraic curve defined by the equation X +Y = 1. The curve Cn is called the Fermat curve. Both the Beta function and the Jacobi sum indeed come from the cohomology of Cn. Let us first consider the case of the Beta function. Let X be an algebraic variety over Q. Between the singular cohomology and the de Rham cohomol- q q ogy, the comparison isomorphism H (X(C), Q) ⊗Q C → HdR(X/Q) ⊗Q C is defined by the so-called period integrals. The Beta function B(s, t)isa 1 period integral for H (Cn)ifs and t are rational numbers with denomina- tor n. Date: April 4, 2002. 1 2 TAKESHI SAITO On the other hand, if p does not divide n, the number of Fp-rational points on Cn is expressed in terms of the Jacobi sums J(a, b) (see [41]). The number of rational points on a variety over a finite field is related to its cohomology. As many readers may already know, the ´etale cohomology was defined by Grothendieck[2] in order to prove the Weil conjecture on the congruence zeta-function, and it was indeed proved using ´etale cohomology, see [12]. The congruence zeta-function is defined as a generating function of the number of Fpm -rational points on the variety. The main point of the proof is to express the congruence zeta-function as the characteristic polynomial of the Frobenius operator acting on the ´etale cohomology. This is how the Jacobi sum J(a, b) is related to the cohomology of the Fermat curve Cn. The cohomology of the Fermat curve Cn is thus related to the Beta func- tion when we regard Q as a subfield of the complex number field C, and to the Jacobi sums when we take a reduction modulo a prime number p. This picture is the prototype of the local-global relation in modern number theory. Beta function C 1 Fermat curve Cn =⇒ H (Cn) p Jacobi sum (Global) (Local) 1.2. -adic representations of the Galois group. Let S be the set of all prime numbers together with the infinity symbol “∞”; i.e., S = {prime numbers} {∞} = {2, 3, 5, 7,... ,∞}. Our standard point of view is to regard S as something analogous to an algebraic curve, and Q as its function field. As each prime number p gives rise to an embedding of Q into the p- adic number field Qp, we associate ∞ to the embedding of Q into R = Q∞. The embeddings corresponding to prime numbers are called finite places, and the embedding of Q into R is called the infinite place. A principle in modern number theory is that the finite places and the infinite place should be treated completely equally. According to this point of view, a locally constant sheaf, or a local system on S (more precisely on an open set of S) is an -adic representation of the absolute Galois group GQ. In Topology a local system on a topological space S is nothing but a rep- resentation of its fundamental group π1(S). The fundamental group π1(S) controls the coverings of the space S. Since the absolute Galois group GQ controls the coverings of (an open set of) S = {2, 3, 5, 7,... ,∞}, we consider a representation of GQ as a local system on (an open set of) S. Recall that the absolute Galois group GQ = Gal(Q¯ /Q) is the automor- phism group Aut(Q¯ ) of an algebraic closure Q¯ of Q.If is a prime num- ber, an -adic representation of GQ is a continuous representation GQ → GALOIS REPRESENTATIONS IN ARITHMETIC GEOMETRY II 3 GLQ (V ) GLn(Q), where V is an n-dimensional vector space over the - adic number field Q. It is customary to use p for a prime number regarded as a point in S, and for the coefficient field of the local system. We need to consider Hodge structures together with -adic representations in order to treat the infinite place equally (see [33]). To an algebraic variety or a modular form over Q, we can associate an -adic representation (and a Hodge structure): algebraic variety, -adic representation =⇒ modular form (+ Hodge structure) Such a correspondence enables us to study algebraic geometric objects or representation theoretic objects using -adic representations, which is a lin- ear algebraic object. Conversely, we can study Galois representation, a highly arithmetic object, using geometric or representation theoretic meth- ods. If we take the Fermat curve as the algebraic variety above, we obtain the example given at the beginning. In practice, ´etale cohomology is a prin- cipal tool to construct Galois representations. Though there are some other methods, including that using fundamental groups or congruence, we do not touch these here. Let X be an algebraic variety over Q, and a prime number. Then, the q natural action of GQ on H (XQ¯ , Q) gives rise to an -adic representation q of GQ.AsaQ-vector space, H (XQ¯ , Q) is identified with the coefficient q extension H (X(C), Q) ⊗Q Q, if we fix an embedding of Q¯ into C. The corresponding Hodge structure is defined by the comparison isomorphism q q H (X(C), Q) ⊗Q C → HdR(X/Q) ⊗Q C of the singular cohomology with the algebraic de Rham cohomology. Example 1. Let E be an elliptic curve over Q. The Tate module TE = n ¯ ¯ ←lim− n Ker( : E(Q) → E(Q)) is a free Z-module of rank 2, and it admits a natural representation of the Galois group Gal(Q¯ /Q). The -adic repre- 1 sentation H (EQ¯ , Q) is naturally identified with the dual Hom(TE,Q). If we regard E(C) as the quotient C/T by a lattice T ⊂ C, then we have 1 TE T ⊗Z Z, and H (EQ¯ , Q) Hom(T,Q). ∞ n Example 2. Let f = q=1 anq be a cusp form that is a simultaneous eigenvector of all the Hecke operators Tn. Then we can construct a two- dimensional -adic representation Vf of the Galois group Gal(Q¯ /Q) such that at almost all prime numbers p, it is unramified and Tr(Frp : Vf )=ap (see [7]). This is called the -adic representation associated to the modular form f. Many readers may know that Fermat’s Last Theorem was proved recently, using such Galois representations (see [42]). It was in fact proved by show- ing that the -adic representation in Example 1 is isomorphic to the -adic representation associated to a modular form as in Example 2 (cf.[29]). 4 TAKESHI SAITO 1.3. The congruence zeta-function and Galois representations. A standard method for studying a local system on S = {2, 3, 5, 7,... ,∞} is to study the restriction at each point of S. We may consider each point of S as something similar to a circle S1, which has a nontrivial fundamental group. The reason is that a point of S is a prime number p and its fundamental group is the absolute Galois group GFp . The absolute Galois group GFp = Gal(F¯ p/Fp) is the automorphism group Aut(F¯ p) of an algebraic closure F¯ p of Fp. The p-th power map ϕp : F¯ p → F¯ p is a topological generator of GFp . We call its inverse the geometric Frobenius isomorphism, and denote it by Frp. From now on we identify the absolute Galois group GFp with ˆ ˆ the profinite completion Z of Z via the isomorphism Z → GFp :1→ Fr p. Each point of S has the completion of Z as its fundamental group, and it resembles S1. By Chebotarev’s density theorem a local system of S is more or less determined by its restriction at each point of S. Thus, to determine the restrictions of a local system on S at points of S is an important step in its study. Let X be an n-dimensional non-singular projective variety over Q. In or- der to determine the -adic cohomology of X at each prime p, it often suffices to study the reduction, X mod p. As we will see soon, for almost all p the congruence zeta-function of X mod p determines the -adic representation at p.
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